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Multiscroll in Coupled Double Scroll Type Oscillators

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Multiscroll in Coupled Double Scroll Type Oscillators MULTISCROLL IN COUPLED DOUBLE SCROLL TYPE OSCILLATORS SYAMAL KUMAR DANA Central Instrumentation, Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India [email protected] BRAJENDRA K. SINGH Department of Infectious Disease Epidemiology, Imperial College London, St Mary's Campus, London W2 1PG, UK [email protected] SATYABRATA.CHAKRABORTY1, RAM CHANDRA YADAV2 Central Instrumentation, Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India [email protected], [email protected] JÜRGEN KURTHS Institute of Physics, University of Potsdam, Am Neuen Palais, D-14415 Potsdam, Germany [email protected] GREGORY V. OSIPOV Faculty of Computational Mathematics and Cybernetics, University of Nizhny Novgorod, 603950 Nizhny Novogorod, Russia [email protected] PRODYOT KUMAR ROY Department of Physics, Presidency College, Kolkata 700073, India [email protected] CHIN-KUN HU Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan and Center for Nonlinear and Complex Systems and Department of Physics, Chung-Yuan Christian University, Chungli 32023, Taiwan [email protected] Abstract: A unidirectional coupling scheme is investigated in double scroll type chaotic oscillators that reveal interesting multiscroll dynamics. Instead of using self-oscillatory systems, in this scheme, double scroll chaos from one oscillator is forced into another similar oscillator in a resting state. This coupling scheme is explored in the Chua oscillator, a modified Chua oscillator and the Lorenz oscillator. We have modified the Chua oscillator by simply changing its piecewise linear function a little bit and thereby derived a new 3-scroll attractor. We have observed 4-scroll, 6-scroll attractors in the driven Chua oscillator and the modified Chua oscillator respectively in an intermittency regime of weaker coupling. We have extended the coupling scheme to the Lorenz system when even more interesting multiscroll dynamics (3-, 4-, 5-, 6–scroll) is observed with decreasing coupling strength. It appears as if a hidden multiscroll structure unfolds with weakening coupling interactions. One after another additional scroll appears in the driven Lorenz system when the coupling strength is gradually decreased in the weaker coupling regime. The origin of such multiscroll dynamics is explained using eigenvalue analysis and a bifurcation diagram. A schematic diagram of the multiscroll trajectories is presented to further elucidate the evolution of the scrolls. Experimental evidences are also presented using the Chua circuit and an electronic analog of the Lorenz system. 1. Introduction Studies on multiscroll chaos are made [Lü and Chen, 2006 and refs. therein], in recent times, in search of suitable hyperchaotic circuits for applications in secure communication and in other chaos-based information technologies. The basis of such multiscroll dynamics is mostly double scroll systems like the Chua oscillator [Chua, Komuro and Matsumoto, 1986]. The Chua oscillator has a piecewise linear function with three linear parts and two breakpoints which is an essential criterion for double scroll dynamics. The trajectory of double scroll in the Chua oscillator rotates around either of two saddle foci symmetric to a saddle focus at the origin. The trajectory switches irregularly from one to the other symmetric locations but always repelled by the third saddle focus origin. Multiscroll attractors have a larger number of scrolls (n>2, n is the number of scroll) and mostly been implemented [Kapitaniak et al, 1990; Kapitaniak et al, 1994; Suykens et al, 1993; Arena et al, 1996; Tang et al, 2001; Yalcin et al, 2000; Yalcin et al, 2001; Lü et al, 2003; Ozoguz et al, 2002; Cafagna et al, 2003; Lü et al, 2004; Lü et al, 2004a; Yu et al, 2006; Lü et al, 2006] by introducing additional saddle foci in terms of added breakpoints in the model system. Multiscroll has been reported in unidirectionally or diffusively coupled self-oscillating Chua circuits [Kapitaniak and 1 Chua, 1994] and in arrays of 1-D cellular neural networks (CNN) [Suykens and Vandewalle, 1993], where the birth of a double-double scroll attractor is found in intermittent bursts from 3D manifold to higher dimensions for weaker coupling. However, in such attempts too, for the generation of multiscroll, additional break point was necessitated in the piecewise linear function of the Chua circuit. The electronic implementation of multiscroll is done with different circuit schemes by adding multiple breakpoints in the piecewise linear function of the system [Kapitaniak et al, 1994; Suykens et al, 1993; Arena et al, 1996; Suykens et al, 1997; Yalcin et al, 2000; Yalcin et al, 2001, Cafagna et al, 2003], by using a sine function [Tang et al, 2001] or by adding saturated function series [Lü et al, 2004]. Hysteresis [Lü et al, 2004a] is also used sometimes for the generation of n-scroll chaos. The important practical applications of multiscroll are found as broadband signal generator and as true pseudorandom number generator for communication engineering. The Lorenz system is another class of double scroll system where the nonlinearity exists not in the form of any piecewise linear function but in a straightforward manner in the governing system equations. The trajectory of a typical butterfly attractor in the Lorenz system rotates most of the time around either of its two symmetric saddle foci and always repelled by the saddle origin. Some attempts [Lü and Chen, 2006, Yu et al, 2006] have been recently made to generate multiscroll using the Lorenz system. A general multiscroll Lorenz system family has been recently introduced [Yu et al, 2006] by using a nonparametric polynomial transformation in the Lorenz system. A self-feedback control algorithm [Singh and Hu, 2005] has also been found to work nicely to generate multiscroll in a single Lorenz system. In this paper, our main motivation is to investigate the interaction of a chaotic oscillator with another similar oscillator in a resting state using double scroll systems like the Chua oscillator and the Lorenz system and thereby to explore the multiscroll dynamics. In most of the earlier studies of multiscroll dynamics in coupled systems and also on synchronization studies [Osipov, Kurths and Zhou, 2007; Pikovsky, Rosenblum and Kurths, 2000, 2001; Pecora and Carroll, 1995], in general, the major thrust was given on two or more interacting self-oscillating systems . Instead, we stress here on two coupled nonlinear systems where one is oscillatory and intercepting unidirectionally another system in a resting state. However, similar to earlier results [Kapitaniak and Chua, 1994; Suykens and Chua, 1997] in self-oscillating coupled systems, we observed 2n-scroll (n=2-, 3-scroll) chaos in intermittent bursts in both double scroll (n=2) Chua oscillator and 3-scroll (n=3) modified Chua oscillator. Our objective is to show how an n-scroll (n=2, 3,…) system can generate a 2n-scroll attractor when an oscillatory system interact with a resting system via unidirectional coupling. We simulate a new 3-scroll system for this purpose. In fact, a 3-scroll attractor is reported earlier using either a sine function [Tang et al, 2001] or a voltage controlled current source [Yalcin, Suykens and Vandawalle, 2000] in the Chua circuit. Instead, we derive the new 3-scroll attractor simply by making a small change in the piecewise linear function of the Chua oscillator. Next, we extend the unidirectional coupling scheme to Lorenz system and are able to generate interesting multiscroll dynamics as if a complex hidden geometry gradually unfolds with decreasing coupling interaction. One after another scroll (3-, 4-, 5-, 6-scroll) is added to the driven Lorenz system when the coupling interaction is decreased. We confirm our numerical results with electronic experiments using the Chua circuit and an analog Lorenz circuit. The paper is structured as follows. In section 2, we present the numerical results of the multiscroll dynamics using Chua oscillator, modified Chua oscillator and present the electronic experiment using two coupled Chua circuit. In section 3, we describe our numerical and experimental results in coupled Lorenz system. We explained the origin of multiscroll in the coupled Lorenz system by using eigenvalue analysis and by revealing the bifurcation scenario. We present a schematic diagram of the multiscroll trajectories in coupled Lorenz system to elucidate their evolution in the context of stability of the equilibrium points. Finally we summarize the results in section 4. 2. Multiscroll in coupled Chua's Circuit Coupled systems are usually investigated from the viewpoint of synchronization [Osipov, Kurths and Zhou, 2007; Pikovsky, Rosenblum and Kurths, 2001] which involves two or more self-oscillatory systems . In exp loring multiscroll dynamics too, self-oscillatory chaotic systems are studied using drive-response type unidirectional coupling in the Chua circuit [Kapitaniak and Chua, 1994] or CNNs [Suykens and Chua, 1997] with added breakpoints. An intermittency regime is reported in such coupled self-oscillatory double scroll systems . In this intermittency regime of coupled double scroll or 2-scroll oscillators, a 4-scroll attractor is really born when the double scroll attractor shows intermittent jumps to another double scroll in higher dimension. Instead of using two self-oscillatory systems , here we use two oscillators, one in an oscillatory state and the other one in a resting state. In other words, the double scroll chaos from one
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